Analysis of nuclear structure in a converging expansion scheme

TL;DR

This paper investigates the convergence of a Fermi momentum expansion scheme in nuclear energy density functionals, validating the minimal parameters needed for accurate descriptions of nuclear matter and finite nuclei.

Contribution

It demonstrates that only a few parameters are sufficient for accurate nuclear matter and finite nuclei modeling within the expansion scheme, showing higher-order terms are unnecessary.

Findings

Symmetric nuclear matter is well described with 3 parameters.

Symmetry energy requires 4 parameters for accurate modeling.

Higher-order derivatives do not significantly improve finite nuclei descriptions.

Abstract

In the framework of the KIDS generalized energy density functional (EDF), the nuclear equation of state (EoS) is expressed as an expansion in powers of the Fermi momentum or the cubic root of the density ($\rho^{1/3}$). Although an optimal number of converging terms was obtained in specific cases of fits to empirical data and pseudodata, the degree of convergence remains to be examined not only for homogeneous matter but also for finite nuclei. One goal of the present work is to validate the minimal and optimal number of EoS parameters required for the description of homogeneous nuclear matter over a wide range of densities relevant for astrophysical applications. The major goal is to examine the validity of the adopted expansion scheme for an accurate description of finite nuclei. To this end we vary the values of the high-order derivatives of the EoS, namely the skewness of the energy of symmetric nuclear matter and the kurtosis of the symmetry energy, at saturation and examine the relative importance of each term in $\rho^{1/3}$ expansion for homogeneous matter. For given sets of EoS parameters determined in this way, we define equivalent Skyrme-type functionals and examine the convergence in the description of finite nuclei focusing on the masses and charge radii of closed-shell nuclei. The EoS of symmetric nuclear matter is found to be efficiently parameterized with only 3 parameters and the symmetry energy (or the energy of pure neutron matter) with 4 parameters when the EoS is expanded in the power series of the Fermi momentum. Higher-order EoS parameters do not produce any improvement, in practice, in the description of nuclear ground-state energies and charge radii, which means that they cannot be constrained by bulk properties of nuclei.

Figures

Tables

Table 1. Table 1: Fitted values of parameters α i subscript 𝛼 𝑖 \alpha_{i} in units of MeV ⋅ fm 3 + i ⋅ MeV superscript fm 3 𝑖 $\mathrm{M}\mathrm{e}\mathrm{V}$\cdot$\mathrm{f}\mathrm{m}$^{3+i} . Model S3b with N = 3 𝑁 3 N=3 the EoS parameters are fixed assuming α 3 = 0 subscript 𝛼 3 0 \alpha_{3}=0 with ρ 0 = 0.16 subscript 𝜌 0 0.16 \rho_{0}=0.16  fm -3 , E 0 = − 16.0 subscript 𝐸 0 16.0 E_{0}=-16.0  MeV, and K 0 = 240.0 subscript 𝐾 0 240.0 K_{0}=240.0  MeV with β i subscript 𝛽 𝑖 \beta_{i} of KIDS-ad2. Models S4a, S4b, and S4c correspond to Q 0 = − 360 subscript 𝑄 0 360 Q_{0}=-360 , − 390 390 -390 , and − 420 420 -420  MeV, respectively. For S3b, we obtain Q 0 = − 372.65 subscript 𝑄 0 372.65 Q_{0}=-372.65  MeV. The EoS of PNM is fixed by the baseline parameters shown at the bottom, which corresponds to KIDS-ad2.

Model	$N$	${\alpha }_0$	${\alpha }_1$	${\alpha }_2$	${\alpha }_3$
S3b	3	$-664.52$	$763.55$	$40.13$	$0$
S4a	4	$-677.69$	$836.34$	$-93.95$	$82.33$
S4b	4	$-646.44$	$663.65$	$224.15$	$-112.99$
S4c	4	$-615.19$	$490.96$	$542.24$	$-308.30$
PNM	$N$	${\alpha }_0+{\beta }_0$	${\alpha }_1+{\beta }_1$	${\alpha }_2+{\beta }_2$	${\alpha }_3+{\beta }_3$
KIDS-ad2	4	$-411.13$	$1007.78$	$-1354.64$	$956.47$

Table 2. Table 2: Binding energies per nucleon and charge radii of selected spherical magic nuclei computed with four EoS parameter sets. Top three values with an asterisk for \nuclide [40]Ca, \nuclide [48]Ca, and \nuclide [208]Pb represent input data and the others are predictions. Numbers in parentheses denote the percentage deviations of predictions from data. Experimental data are from Refs.   NNDC ; AM13 .

Nucleus	Binding energy per nucleon $\left(E/A\right)$ (MeV)					Charge radius $\left(R_c\right)$ (fm)
	Expt.	S3b	S4a	S4b	S4c	Expt.	S3b	S4a	S4b	S4c
\nuclide[40]Ca	${8.5513}^{\ast }$	$8.5565$	$8.5579$	$8.5544$	$8.5512$	${3.4776}^{\ast }$	$3.4781$	$3.4799$	$3.4758$	$3.4720$
		$\left(0.060\%\right)$	$\left(0.078\%\right)$	$\left(0.037\%\right)$	$\left(0.001\%\right)$		$\left(0.014\%\right)$	$\left(0.066\%\right)$	$\left(0.052\%\right)$	$\left(0.161\%\right)$
\nuclide[48]Ca	${8.6667}^{\ast }$	$8.6564$	$8.6569$	$8.6558$	$8.6549$	${3.4771}^{\ast }$	$3.4867$	$3.4882$	$3.4847$	$3.4813$
		$\left(0.120\%\right)$	$\left(0.113\%\right)$	$\left(0.126\%\right)$	$\left(0.136\%\right)$		$\left(0.277\%\right)$	$\left(0.319\%\right)$	$\left(0.220\%\right)$	$\left(0.122\%\right)$
\nuclide[208]Pb	${7.8675}^{\ast }$	$7.8809$	$7.8816$	$7.8800$	$7.8783$	${5.5012}^{\ast }$	$5.4887$	$5.4901$	$5.4870$	$5.4840$
		$\left(0.172\%\right)$	$\left(0.179\%\right)$	$\left(0.160\%\right)$	$\left(0.138\%\right)$		$\left(0.228\%\right)$	$\left(0.201\%\right)$	$\left(0.259\%\right)$	$\left(0.313\%\right)$
\nuclide[16]O	$7.9762$	$7.8684$	$7.8675$	$7.8686$	$7.8678$	$2.6991$	$2.7618$	$2.7643$	$2.7587$	$2.7541$
		$\left(1.35\%\right)$	$\left(1.36\%\right)$	$\left(1.35\%\right)$	$\left(1.36\%\right)$		$\left(2.322\%\right)$	$\left(2.41\%\right)$	$\left(2.209\%\right)$	$\left(2.036\%\right)$
28O	—	$6.0646$	$6.0640$	$6.0650$	$6.0649$	—	$2.8371$	$2.8384$	$2.8351$	$2.8315$
60Ca	—	$7.6561$	$7.6567$	$7.6552$	$7.6535$	—	$3.6465$	$3.6478$	$3.6445$	$3.6411$
\nuclide[90]Zr	$8.7100$	$8.7328$	$8.7345$	$8.7309$	$8.7282$	$4.2694$	$4.2476$	$4.2488$	$4.2459$	$4.2428$
		$\left(0.263\%\right)$	$\left(0.281\%\right)$	$\left(0.241\%\right)$	$\left(0.209\%\right)$		$\left(0.510\%\right)$	$\left(0.482\%\right)$	$\left(0.550\%\right)$	$\left(0.622\%\right)$
\nuclide[132]Sn	$8.3549$	$8.3563$	$8.3559$	$8.3565$	$8.3565$	$4.7093$	$4.7089$	$4.7100$	$4.7072$	$4.7044$
		$\left(0.017\%\right)$	$\left(0.013\%\right)$	$\left(0.020\%\right)$	$\left(0.020\%\right)$		$\left(0.009\%\right)$	$\left(0.015\%\right)$	$\left(0.044\%\right)$	$\left(0.103\%\right)$

Table 3. Table 3: Fitted parameters of Skyrme functional parameters. Here, we set y 1 = y 2 = 0 subscript 𝑦 1 subscript 𝑦 2 0 y_{1}=y_{2}=0 and ζ 𝜁 \zeta is dimensionless.

Parameter	S3b	S4a	S4b	S4c
$t_0$ ($\text{MeV}\cdot {\text{fm}}^3$)	$-1772.04$	$-1807.17$	$-1723.84$	$-1640.50$
$y_0$ ($\text{MeV}\cdot {\text{fm}}^3$)	$-127.52$	$-162.65$	$-79.32$	$4.02$
$t_1$ ($\text{MeV}\cdot {\text{fm}}^5$)	$275.72$	$262.17$	$288.94$	$303.28$
$t_2$ ($\text{MeV}\cdot {\text{fm}}^5$)	$-161.50$	$-167.94$	$-154.90$	$-146.98$
$t_{31}$ (${10}^4\text{MeV}\cdot {\text{fm}}^4$)	$1.222$	$1.338$	$1.062$	$0.7855$
$y_{31}$ (${10}^4\text{MeV}\cdot {\text{fm}}^4$)	$-1.197$	$-1.081$	$-1.357$	$-1.633$
$t_{32}$ ($\text{MeV}\cdot {\text{fm}}^5$)	$571.0$	$-1310.7$	$3252.4$	$8043.0$
$y_{32}$ (${10}^4\text{MeV}\cdot {\text{fm}}^5$)	$2.949$	$2.704$	$3.274$	$3.818$
$t_{33}$ ($\text{MeV}\cdot {\text{fm}}^6$)	—	$1317.2$	$-1807.8$	$-4932.8$
$y_{33}$ (${10}^4\text{MeV}\cdot {\text{fm}}^6$)	$-2.296$	$-2.164$	$-2.476$	$-2.789$
$\zeta$	0.1106	0.1281	0.0931	0.0729
$W_0$ ($\text{MeV}\cdot {\text{fm}}^5$)	108.35	106.79	109.88	111.55

Table 4. Table 4: Values of c i ​ ( 1 ) subscript 𝑐 𝑖 1 c_{i}(1) fitted to APR EoS of PNM. The unit of c i subscript 𝑐 𝑖 c_{i} is MeV ⋅ fm 3 + i ⋅ MeV superscript fm 3 𝑖 $\mathrm{M}\mathrm{e}\mathrm{V}$\cdot$\mathrm{f}\mathrm{m}$^{3+i} and the units of J 𝐽 J , L 𝐿 L , K sym subscript 𝐾 sym K_{\rm sym} , Q sym subscript 𝑄 sym Q_{\rm sym} , and R sym subscript 𝑅 sym R_{\rm sym} are MeV.

Model	$N$	$c_0\left(1\right)$	$c_1\left(1\right)$	$c_2\left(1\right)$	$c_3\left(1\right)$	$c_4\left(1\right)$	$c_5\left(1\right)$	${\chi }_n^2$	$J$	$L$	$K_{\mathrm{sym}}$	$Q_{\mathrm{sym}}$	$R_{\mathrm{sym}}$
P3	3	$-266.72$	$133.50$	$281.38$	—	—	—	$5.3\times {10}^{-4}$	$32.6$	$53.5$	$-129.7$	$422.3$	$-2421.8$
P4	4	$-407.94$	$990.09$	$-1321.86$	$937.14$	—	—	$1.4\times {10}^{-4}$	$32.8$	$49.2$	$-156.3$	$583.1$	$-2469.7$
P5	5	$-224.16$	$-479.28$	$2814.48$	$-3963.71$	$2075.79$	—	$6.3\times {10}^{-5}$	$33.0$	$51.4$	$-166.8$	$461.4$	$-1388.4$
P6a	6	$-224.81$	$-473.46$	$2795.50$	$-3935.18$	$2056.11$	$4.94$	$6.3\times {10}^{-5}$	$33.0$	$51.4$	$-166.8$	$461.6$	$-1391.7$
P6b	6	$-283.99$	$110.63$	$604.05$	$-10.59$	$-1312.44$	$1117.76$	$6.4\times {10}^{-5}$	$33.0$	$51.5$	$-163.8$	$450.0$	$-1545.9$
P6c	6	$-313.98$	$400.88$	$-463.41$	$1864.00$	$-2891.61$	$1630.37$	$6.5\times {10}^{-5}$	$33.0$	$51.5$	$-162.3$	$446.6$	$-1631.2$

Table 5. Table 5: Same as Table  4 but for QMC EoS of PNM of Ref.  CGPP14 .

Model	$N$	$c_0\left(1\right)$	$c_1\left(1\right)$	$c_2\left(1\right)$	$c_3\left(1\right)$	$c_4\left(1\right)$	$c_5\left(1\right)$	${\chi }_n^2$	$J$	$L$	$K_{\mathrm{sym}}$	$Q_{\mathrm{sym}}$	$R_{\mathrm{sym}}$
QMC P3	3	$-119.01$	$-424.80$	$841.72$	—	—	—	$1.4\times {10}^{-4}$	$34.1$	$62.5$	$-59.5$	$566.0$	$-3304.7$
QMC P4	4	$-395.79$	$1085.08$	$-1818.25$	$1519.80$	—	—	$1.5\times {10}^{-6}$	$34.5$	$60.5$	$-88.9$	$751.2$	$-3075.9$
QMC P5	5	$-349.56$	$740.06$	$-876.85$	$404.60$	$484.89$	—	$1.3\times {10}^{-6}$	$34.5$	$60.7$	$-90.0$	$735.7$	$-2876.4$
QMC P6	6	$-293.93$	$238.03$	$896.89$	$-2667.81$	$3098.48$	$-874.88$	$1.3\times {10}^{-6}$	$34.5$	$60.7$	$-90.8$	$737.1$	$-2781.0$

Table 6. Table 6: Same as Table  3 but for P3, P4, and P5. Note that t 33 = t 34 = 0 subscript 𝑡 33 subscript 𝑡 34 0 t_{33}=t_{34}=0 as we use S3b for α i = c i ​ ( 0 ) subscript 𝛼 𝑖 subscript 𝑐 𝑖 0 \alpha_{i}=c_{i}(0) .

Parameter	P3	P4	P5
$t_0$ ($\text{MeV}\cdot {\text{fm}}^3$)	$-1772.04$	$-1772.04$	$-1772.04$
$y_0$ ($\text{MeV}\cdot {\text{fm}}^3$)	$-705.16$	$-140.27$	$-875.42$
$t_1$ ($\text{MeV}\cdot {\text{fm}}^5$)	$247.33$	$275.83$	$269.90$
$t_2$ ($\text{MeV}\cdot {\text{fm}}^5$)	$-173.00$	$-161.48$	$-163.95$
$t_{31}$ (${10}^4\text{MeV}\cdot {\text{fm}}^4$)	$12216.73$	$12216.73$	$12216.73$
$y_{31}$ (${10}^4\text{MeV}\cdot {\text{fm}}^4$)	$9012.81$	$-11545.41$	$23719.36$
$t_{32}$ ($\text{MeV}\cdot {\text{fm}}^5$)	$1087.14$	$569.38$	$678.46$
$y_{32}$ (${10}^4\text{MeV}\cdot {\text{fm}}^5$)	$-10346.18$	$28700.54$	$-70692.70$
$y_{33}$ (${10}^4\text{MeV}\cdot {\text{fm}}^6$)	—	$-22491.36$	$95128.93$
$y_{34}$ (${10}^4\text{MeV}\cdot {\text{fm}}^7$)	—	—	$-49818.87$
$\zeta$	$-0.6931$	$0.1133$	$-0.0566$
$W_0$ ($\text{MeV}\cdot {\text{fm}}^5$)	$104.12$	$108.46$	$108.25$

Table 7. Table 7: Same as Table  2 but for P3, P4, and P5. The SNM parameters are fixed to the values of model S3b in Table  1 . The experimental data are from Refs.  NNDC ; AM13 .

Nuclei	Energy per particle (MeV)				Charge radius (fm)
	Expt.	P3	P4	P5	Expt.	P3	P4	P5
40Ca	${8.5513}^{\ast }$	$8.5573$	$8.5564$	$8.5561$	${3.4776}^{\ast }$	$3.4785$	$3.4781$	$3.4782$
		$\left(0.070\%\right)$	$\left(0.059\%\right)$	$\left(0.056\%\right)$		$\left(0.026\%\right)$	$\left(0.014\%\right)$	$\left(0.015\%\right)$
48Ca	${8.6667}^{\ast }$	$8.6556$	$8.6565$	$8.6581$	${3.4771}^{\ast }$	$3.4891$	$3.4867$	$3.4870$
		$\left(0.129\%\right)$	$\left(0.118\%\right)$	$\left(0.099\%\right)$		$\left(0.345\%\right)$	$\left(0.277\%\right)$	$\left(0.285\%\right)$
208Pb	${7.8675}^{\ast }$	$7.8849$	$7.8806$	$7.8793$	${5.5012}^{\ast }$	$5.4934$	$5.4886$	$5.4891$
		$\left(0.222\%\right)$	$\left(0.167\%\right)$	$\left(0.151\%\right)$		$\left(0.141\%\right)$	$\left(0.228\%\right)$	$\left(0.221\%\right)$
16O	$7.9762$	$7.8641$	$7.8683$	$7.8669$	$2.6991$	$2.7634$	$2.7618$	$2.7621$
		$\left(1.405\%\right)$	$\left(1.353\%\right)$	$\left(1.371\%\right)$		$\left(2.382\%\right)$	$\left(2.322\%\right)$	$\left(2.335\%\right)$
28O	—	$6.0705$	$6.0628$	$6.0585$	—	$2.8435$	$2.8371$	$2.8396$
60Ca	—	$7.6659$	$7.6548$	$7.6513$	—	$3.6511$	$3.6465$	$3.6478$
90Zr	$8.7100$	$8.7336$	$8.7330$	$8.7344$	$4.2694$	$4.2489$	$4.2476$	$4.2476$
		$\left(0.272\%\right)$	$\left(0.264\%\right)$	$\left(0.280\%\right)$		$\left(0.480\%\right)$	$\left(0.510\%\right)$	$\left(0.511\%\right)$
132Sn	$8.3549$	$8.3592$	$8.3559$	$8.3549$	$4.7093$	$4.7133$	$4.7088$	$4.7090$
		$\left(0.052\%\right)$	$\left(0.013\%\right)$	$\left(0.001\%\right)$		$\left(0.085\%\right)$	$\left(0.010\%\right)$	$\left(0.006\%\right)$

Table 8. Table 8: Values of c i ​ ( δ ) subscript 𝑐 𝑖 𝛿 c_{i}(\delta) with δ = 1 𝛿 1 \delta=1 fitted to the symmetry energy parameters J = 32.78 𝐽 32.78 J=32.78  MeV, L = 49.25 𝐿 49.25 L=49.25  MeV, K sym = − 156.26 subscript 𝐾 sym 156.26 K_{\rm sym}=-156.26  MeV, Q sym = 583.07 subscript 𝑄 sym 583.07 Q_{\rm sym}=583.07  MeV, and three R sym subscript 𝑅 sym R_{\rm sym} values. The unit of R sym subscript 𝑅 sym R_{\rm sym} is MeV and c i subscript 𝑐 𝑖 c_{i} is in the unit of MeV ⋅ fm 3 + i ⋅ MeV superscript fm 3 𝑖 $\mathrm{M}\mathrm{e}\mathrm{V}$\cdot$\mathrm{f}\mathrm{m}$^{3+i} .

Model	$R_{\mathrm{sym}}$	$c_0\left(1\right)$	$c_1\left(1\right)$	$c_2\left(1\right)$	$c_3\left(1\right)$	$c_4\left(1\right)$
P5a	$-2170$	$-329.19$	$411.12$	$275.64$	$-1022.73$	$901.92$
P5b	$-2470$	$-407.32$	$986.75$	$-1314.84$	$930.40$	$2.50$
P5c	$-2770$	$-485.44$	$1562.38$	$-2905.32$	$2883.52$	$-896.92$

Table 9. Table 9: Same as Table  4 but for P5a, P5b, and P5c. Here, t 33 = t 34 = 0 subscript 𝑡 33 subscript 𝑡 34 0 t_{33}=t_{34}=0 as we use S3b for α i = c i ​ ( 0 ) subscript 𝛼 𝑖 subscript 𝑐 𝑖 0 \alpha_{i}=c_{i}(0)

Parameter	P5a	P5b	P5c
$t_0$ ($\text{MeV}\cdot {\text{fm}}^3$)	$-1772.04$	$-1772.04$	$-1772.04$
$y_0$ ($\text{MeV}\cdot {\text{fm}}^3$)	$-455.27$	$-142.77$	$169.73$
$t_1$ ($\text{MeV}\cdot {\text{fm}}^5$)	$246.82$	$275.85$	$273.68$
$t_2$ ($\text{MeV}\cdot {\text{fm}}^5$)	$-173.35$	$-161.47$	$-162.34$
$t_{31}$ (${10}^4\text{MeV}\cdot {\text{fm}}^4$)	$12216.73$	$12216.73$	$12216.73$
$y_{31}$ (${10}^4\text{MeV}\cdot {\text{fm}}^4$)	$2349.81$	$-11465.31$	$-25280.43$
$t_{32}$ ($\text{MeV}\cdot {\text{fm}}^5$)	$1099.03$	$569.01$	$608.27$
$y_{32}$ (${10}^4\text{MeV}\cdot {\text{fm}}^5$)	$-10223.47$	$28532.42$	$66660.82$
$y_{33}$ (${10}^4\text{MeV}\cdot {\text{fm}}^6$)	$24545.47$	$-22329.53$	$-69204.53$
$y_{34}$ (${10}^4\text{MeV}\cdot {\text{fm}}^6$)	$-21646.05$	$-59.93$	$21526.20$
$\zeta$	$-0.7116$	$0.1139$	$0.0527$
$W_0$ ($\text{MeV}\cdot {\text{fm}}^5$)	$105.57$	$108.49$	$107.90$

Table 10. Table 10: Same as Table  7 but for P5a, P5b, and P5c. The SNM parameters are fixed to the values of model S3b in Table  1 . The experimental data are from Refs.  NNDC ; AM13 .

Nuclei	Binding energy per nucleon [MeV]				Charge radius [fm]
	Expt.	P5a	P5b	P5c	Expt.	P5a	P5b	P5c
40Ca	${8.5513}^{\ast }$	$8.5567$	$8.5564$	$8.5564$	${3.4776}^{\ast }$	$3.4786$	$3.4781$	$3.4782$
		$\left(0.063\%\right)$	$\left(0.060\%\right)$	$\left(0.060\%\right)$		$\left(0.029\%\right)$	$\left(0.014\%\right)$	$\left(0.018\%\right)$
48Ca	${8.6667}^{\ast }$	$8.6575$	$8.6566$	$8.6560$	${3.4771}^{\ast }$	$3.4872$	$3.4867$	$3.4863$
		$\left(0.106\%\right)$	$\left(0.117\%\right)$	$\left(0.123\%\right)$		$\left(0.291\%\right)$	$\left(0.276\%\right)$	$\left(0.264\%\right)$
208Pb	${7.8675}^{\ast }$	$7.8800$	$7.8806$	$7.8808$	${5.5012}^{\ast }$	$5.4891$	$5.4886$	$5.4880$
		$\left(0.159\%\right)$	$\left(0.167\%\right)$	$\left(0.170\%\right)$		$\left(0.221\%\right)$	$\left(0.229\%\right)$	$\left(0.240\%\right)$
16O	$7.9762$	$7.8633$	$7.8683$	$7.8679$	$2.6991$	$2.7636$	$2.7618$	$2.7619$
		$\left(1.42\%\right)$	$\left(1.35\%\right)$	$\left(1.36\%\right)$		$\left(2.39\%\right)$	$\left(2.32\%\right)$	$\left(2.33\%\right)$
28O	—	$6.0467$	$6.0623$	$6.0746$	—	$2.8381$	$2.8371$	$2.8353$
60Ca	—	$7.6470$	$7.6545$	$7.6611$	—	$3.6475$	$3.6465$	$3.6451$
90Zr	$8.7100$	$8.7357$	$8.7330$	$8.7322$	$4.2694$	$4.2474$	$4.2476$	$4.2474$
		$\left(0.295\%\right)$	$\left(0.265\%\right)$	$\left(0.255\%\right)$		$\left(0.516\%\right)$	$\left(0.511\%\right)$	$\left(0.516\%\right)$
132Sn	$8.3549$	$8.3539$	$8.3559$	$8.3564$	$4.7093$	$4.7093$	$4.7088$	$4.7082$
		$\left(0.012\%\right)$	$\left(0.013\%\right)$	$\left(0.019\%\right)$		$\left(0.000\%\right)$	$\left(0.010\%\right)$	$\left(0.024\%\right)$

Equations

$$\mathcal{E}(\rho ,\delta )=\mathcal{T}(\rho ,\delta )+\sum _{i=0}^{N-1}{c}_i(\delta ){\rho }^{1+i/3},$$

$$c_i(\delta )={\alpha }_i+{\beta }_i{\delta }^2,$$

$$\mathcal{S}(\rho )={\frac{1}{2}\frac{{\partial }^2}{\partial {\delta }^2}\mathcal{E}(\rho ,\delta )|}_{\delta =0}={\mathcal{T}}_{\mathrm{sym}}(\rho )+\sum _{i=0}^{N-1}{\beta }_i{\rho }^{1+i/3},$$

$${\mathcal{T}}_{\mathrm{sym}}(\rho )=\frac{{\hslash }^2}{6m}{\left(\frac{3{\pi }^2}{2}\right)}^{2/3}{\rho }^{2/3}.$$

$$\mathcal{E}(\rho ,0)$$

$$\mathcal{S}(\rho )$$

$$K_0$$

$$Q_0$$

$$L$$

$$K_{\mathrm{sym}}$$

$$Q_{\mathrm{sym}}$$

$$R_{\mathrm{sym}}$$

$$K_0$$

$$Q_0$$

$$K_{\mathrm{sym}}$$

$$Q_{\mathrm{sym}}$$

$$R_{\mathrm{sym}}$$

$$v_{ij}$$

$$\mathcal{E}$$

$$t_0$$

$$t_{3n}$$

$$t_{32}$$

$$y_{32}$$

$${\theta }_s$$

$${\theta }_{\mu }$$

$${\mu }_s^{-1}\equiv (m_{}^{\ast }/m{)}^{-1}$$

$${\mu }_v^{-1}\equiv (m_v^{\ast }/m{)}^{-1}$$

$${\chi }^2=\sum _j\exp \left(-\tilde{\beta }{\rho }_j/{\rho }_0\right){\left(\frac{\mathcal{E}({\rho }_j)-D_j}{\mathcal{T}({\rho }_j)}\right)}^2,$$

$${\chi }_n^2={\chi }^2/\sum _j\exp \left(-\tilde{\beta }{\rho }_j/{\rho }_0\right)$$

Full text

Purpose: The first goal is to validate the minimal and optimal number of EoS parameters required for the description of homogeneous nuclear matter over a wide range of densities relevant for astrophysical applications. The major goal is to examine the validity of the adopted expansion scheme for an accurate description of finite nuclei.

Method: We vary the values of the high-order density derivatives of the nuclear EoS, such as the skewness of the energy of symmetric nuclear matter and the kurtosis of the symmetry energy, at saturation and examine the relative importance of each term in $\rho^{1/3}$ expansion for homogeneous matter. For given sets of EoS parameters determined in this way, we define equivalent Skyrme-type functionals and examine the convergence in the description of finite nuclei focusing on the masses and charge radii of closed-shell nuclei.

Results: The EoS of symmetric nuclear matter is found to be efficiently parameterized with only 3 parameters and the symmetry energy (or the energy of pure neutron matter) with 4 parameters when the EoS is expanded in the power series of the Fermi momentum. Higher-order EoS parameters do not produce any improvement, in practice, in the description of nuclear ground-state energies and charge radii, which means that they cannot be constrained by bulk properties of nuclei.

Conclusions: The minimal nuclear EDF obtained in the present work is found to reasonably describe the properties of closed-shell nuclei and the mass-radius relation of neutron stars. Attempts at refining the nuclear EDF beyond the minimal formula must focus on parameters which are not active (or strongly active) in unpolarized homogeneous matter, for example, effective tensor terms and time-odd terms.

I Introduction

In a series of publication PPLH16 ; GOHP17 ; GPHPO16 ; GPHO18 , we have proposed and developed a strategy to model nuclear systems based on a converging power expansion combined with energy density functional (EDF) theory. Beginning with homogeneous matter PPLH16 , we formulated the energy per particle, which represents the equation of state (EoS), as an expansion in powers of the Fermi momentum or equivalently in powers of the cubic root of the density, as $k_{F}\propto\rho^{1/3}$. This choice is rooted both in quantum many-body theory and effective field theory. We confirmed a posteriori the quick convergence of the expansion by fitting the parameters to pseudodata from microscopic calculations. Based on a statistical analysis of the fits, a robust parameter set was chosen as a baseline for further explorations, comprising three terms for isospin-symmetric nuclear matter (SNM) and four for pure neutron matter (PNM). The naturalness of the expansion was confirmed and extrapolations to extreme density regimes, were found to be satisfactory GPHO18 . In particular, the extrapolated results agreed with ab initio calculations for dilute neutron matter, a regime to which the model had not been fitted at all, and reproduced a realistic mass-radius relation of neutron stars, which represents a dense regime.

In subsequent works reported in Refs. GOHP17 ; GPHPO16 ; GPHO18 , the KIDS EoS was transposed to a Skyrme functional with extended density-dependent couplings, which we call a KIDS EDF, to study nuclear ground-state properties, thereby relying on the Kohn-Sham scheme KS65 ; BHR03 .With the baseline EoS from Ref. PPLH16 and only six input data, namely the ground-state energies and charge radii of three nuclei, it was possible to obtain a successful description of the bulk properties of closed-(sub)shell nuclei over a wide range of atomic number, say from \nuclide[16]O to \nuclide[218]U GOHP17 ; GPHPO16 ; GPHO18 .111Because only closed-(sub)shell nuclei were considered, we do not include pairing interactions in the present work. Furthermore, the results were found to be practically independent of the assumption on the in-medium effective mass GPHO18 , which means that the latter cannot be efficiently constrained by the bulk static properties of nuclei. The corresponding parameters then remain to be determined via dynamic properties of nuclei. The above results showed that with a well-defined nuclear EoS Ansatz, the convenient Skyrme formalism, and simple rules for fitting, it would be possible to find a unified and phenomenological nuclear model describing nuclear matter and nuclei with the same parameter set, i.e., the same EoS.

Before developing more sophisticated models to describe various types of nuclei along this approach, we address the convergence issue in the description of closed-(sub)shell nuclei at the present stage. Throughout the previous publications  PPLH16 ; GOHP17 ; GPHPO16 ; GPHO18 we have shown that the expansion of the EoS as a power series of the Fermi momentum exhibits excellent convergence well above the saturation density GOHP17 . However, careful analyses lead to the observation that the degree of convergence depends on isospin and, as a result, higher order contributions are more important in PNM than in SNM. In fact, in Ref. PPLH16 , it is shown that, while three terms are sufficient for describing SNM in a fast-converging hierarchy, at least four terms are needed to have such behavior for PNM. The origin of this difference is certainly of theoretical interest and requires sophisticated investigations on nuclear dynamics. Although we will not address here the issue on its fundamental origins, it would be important and meaningful to examine the convergence in the description of finite nuclei. This is the major motivation of the present article and the purpose of the this work is to examine the convergence of the power series expansion in the Fermi momentum for the description of finite nuclei.

The nuclear EoS is often represented in terms of parameters defined at the saturation point such as the saturation density $\rho_{0}$, the binding energy per particle at saturation $E_{0}$, the symmetry energy at saturation $J$, the slope parameter $L$, and the compression modulus $K_{0}$. These parameters were used to constrain the nuclear EoS in our previous publications PPLH16 ; GOHP17 ; GPHPO16 ; GPHO18 . However, the role of the parameters that are related to higher derivatives of the EoS with respect to density remains to be explored. These “EoS parameters” can be readily expressed analytically in terms of the KIDS expansion coefficients. The question of how many KIDS parameters are needed for an efficient description of nuclear systems can be rephrased as how many high-order derivatives of the SNM energy and of the symmetry energy are needed. In other words, we also need to examine how many EoS parameters are necessary for an efficient and well-converged description of PNM and nuclear ground states. Furthermore, since higher-order terms in the power series expansion control the behavior of the nuclear EoS at higher densities, higher-order EoS parameters such as the skewness and kurtosis would help in constraining the EoS at higher densities and examining the convergence of the EoS.

Motivated by the above issues, in the present work we address the following questions. In Refs. PPLH16 ; GOHP17 ; GPHPO16 ; GPHO18 , we successfully parameterized the EoS of SNM and PNM by three and four EoS parameters in the considered range of densities. Then it is natural to seek how far the constructed EoS can be applied as a function of density. This is related to the role of higher order EoS parameters and we explore the sensitivity of our EoS to higher-order EoS parameters. Once their role is identified for homogeneous nuclear matter, we investigate the role of higher-order EoS parameters in the description of finite nuclei. To this end, we obtain results for various values of the skewness of the SNM EoS and the kurtosis of the symmetry energy at the saturation point to confirm that such higher order terms hardly play any role. The corollary is that the skewness of the SNM EoS and the symmetry-energy kurtosis cannot be practically constrained by the static properties of nuclei such as masses and radii.

This paper is organized as follows. In Sec. II, we briefly review the formalism of the KIDS EDF and the corresponding Skyrme potentials will be developed. Section III is devoted to the exploration of the uncertainty in the fourth-order term in SNM and the role of the skewness of the SNM EoS is examined. The mass-radius relations of neutron stars are also computed within the models of the present approach. Then, in Sec. IV, we increase the number of terms in the asymmetric part of EDF to investigate the convergence behavior of the model with respect to the kurtosis of the nuclear symmetry energy. In Sec. V, we discuss the results in the context of current efforts to extend the nuclear EDF, in particular, in the form of extended Skyrme functionals with rich momentum dependence and tensor forces. Finally, we summarize and conclude in Sec. VI.

II KIDS EDF: equation of state and corresponding Skyrme functionals

II.1 KIDS equation of state

In the KIDS model for nuclear EDF, the energy per particle in homogeneous nuclear matter is expanded in powers of the Fermi momentum $k_{F}$ or equivalently the cubic root of the baryon density $\rho$. Thus the nuclear EDF in this approach is written as

[TABLE]

where $\mathscr{T}$ is the free Fermi-gas kinetic energy and the potential energy is expanded up to $N$ terms, namely, up to the order of $\rho^{(N+2)/3}$ starting from the $\rho$ term. The isospin asymmetry $\delta$ is defined as $\delta=(\rho_{n}-\rho_{p})/\rho$, where $\rho_{n}$ and $\rho_{p}$ are neutron and proton densities, respectively, which give the total nucleon density $\rho=\rho_{n}+\rho_{p}$. Model parameters $c_{i}(\delta)$ could be expanded in even powers of isospin asymmetry $\delta$. For the purpose of the present work, we adopt the usual quadratic approximation for the isospin-asymmetry dependence of $c_{i}(\delta)$ by writing

[TABLE]

which leads to $\alpha_{i}=c_{i}(0)$ and $\beta_{i}=c_{i}(1)-c_{i}(0)$.

The expansion parameters $c_{i}(\delta)$ can be constrained once the empirical properties of nuclear matter, i.e., EoS parameters, are known. Phenomenologically, these parameters are defined at nuclear saturation density by the series expansion of the SNM energy $\mathscr{E}(\rho,0)$ and nuclear symmetry energy, which can be defined and expressed as

[TABLE]

where the contribution from the free kinetic energy reads

[TABLE]

Then EoS parameters of interest are defined through DLSD12

[TABLE]

where $x=(\rho-\rho_{0})/(3\rho_{0})$.

Therefore, the SNM energy is characterized by the saturation density $\rho_{0}$, the energy per particle at saturation $E_{0}$, the compression modulus $K_{0}$, and the skewness coefficient $Q_{0}$ defined as

[TABLE]

On the other hand, the nuclear symmetry energy is customarily characterized at the saturation point by its value $J=\mathcal{S}(\rho_{0})$, the slope $L$, and the curvature $K_{\mathrm{sym}}$ defined as

[TABLE]

In addition, we consider the skewness $Q_{\mathrm{sym}}$ and the kurtosis $R_{\mathrm{sym}}$, defined via the 3rd and 4th derivatives, respectively, as

[TABLE]

These EoS parameters will be discussed in the parameterization of the KIDS model.

All the above quantities are readily obtained analytically with the help of expressions of Eqs. (1)–(3). Explicitly, we have

[TABLE]

These relations connect the values of the EoS parameter to our model parameters $\alpha_{i}$ and $\beta_{i}$. Once the values of EoS parameters are known, our approach allows us to find the nuclear EoS to the desired order in density. However, most of the above EoS parameters are not known to a satisfactory accuracy and ranges of their values are to be explored.

In Ref. PPLH16 , we determined the baseline KIDS parameter set labeled ‘KIDS-ad2’ in the following way. We began by fitting many possible combinations (of varying order $N$) of KIDS parameters $\alpha_{i}$ and $\beta_{i}$ to the Akmal-Pandharipande-Ravenhall (APR) EoS APR98 . Having concluded that the three lowest-order terms are sufficient for the description of SNM, we set $\alpha_{3}=0$, and determined $\alpha_{0,1,2}$ by widely adopted empirical properties at saturation, namely, $\rho_{0}=0.16$ $\mbox{fm}^{-3}$, $E_{0}=-16$ MeV, and $K_{0}=240$ MeV.(These values are also consistent with the APR EoS.) This model is then found to give the skewness coefficient $Q_{0}\approx-373$ MeV. The coefficients $c_{i}(1)$, or equivalently $\beta_{i}$, were also fitted to the APR EoS for PNM. In this case, we found that at least four terms had to be retained in the KIDS EDF in order to reproduce the APR EoS for PNM. The resulting EDF gives $J=32.8$ MeV, $L=49.3$ MeV, $K_{\rm sym}=-156$ MeV, $Q_{\rm sym}=583$ MeV, and $R_{\rm sym}=-2470$ MeV.

The KIDS-ad2 EoS determined in this way was subsequently transposed into a zero-range, density-dependent effective interaction for nuclei and applied successfully to Hartree-Fock calculations of nuclear ground states of closed-shell nuclei GPHPO16 ; GPHO18 , providing satisfactory results, on a par with generalized Skyrme-type functionals. The question to be addressed at the present work is to examine whether superior results can be obtained with higher-order terms.

II.2 Corresponding Skyrme functionals

In this subsection, we review a simple procedure for applying a given KIDS EoS to the description of finite nuclei, which will be employed in the present work. The Fermi momentum expansion of EDF in Eq. (1) leads to a convenient Skyrme-type effective interaction GPHO18 in the form of

[TABLE]

where $\mathbf{k}=(\bm{\nabla}_{i}-\bm{\nabla}_{j})/(2i)$, $\mathbf{k}^{\prime}=-(\bm{\nabla}_{i}^{\prime}-\bm{\nabla}_{j}^{\prime})/(2i)$, and $P_{\sigma}$ is the spin-exchange operator. Here, $W_{0}$ denotes the strength of the effective spin-orbit coupling, which is not active in homogeneous matter. It, therefore, must be determined from nuclear data. This is similar in form to a generalized Skyrme model proposed in Refs. CBMBD03 ; CBBDM04 ; ADK06 , but the protocol for determining the Skyrme potential parameters is quite different. In the so-called generalized Skyrme potential model, the parameters are determined by some properties of specific nuclei. In our case, however, we will begin with an unchanged EoS and use very few nuclear data for remaining undetermined parameters. We also retain the freedom to have, e.g., $t_{33}=0$ but $y_{33}\neq 0$. The corresponding EDF in terms of the local densities as well as gradient and kinetic terms can be obtained from a standard calculation as

[TABLE]

where $\tau$ denotes the kinetic energy density and ${\mathbf{J}}$ the current density. The sum over $q$ means the summation over isospin, i.e., $q=(n,p)$. Matching the KIDS EDF in Eq. (2) and the Skyrme functional in Eq. (15) leads to the following relations:

[TABLE]

which defines $\zeta$ and $\zeta^{\prime}$ with

[TABLE]

The matching reveals that there are two sources for the $\rho^{5/3}$ term in the EoS which corresponds to $n=2$ in Eq. (14): one from the density-dependent terms in Eq. (14) with the Skyrme parameters $t_{32}$, $y_{32}$, and the other from the momentum-dependent terms in Eq. (14) with the Skyrme parameters $t_{1}$, $t_{2}$, $y_{1}$, $y_{2}$. The partition is encoded in the unknown parameters $\zeta$ and $\zeta^{\prime}$ in Eqs. (16) and (17). Also undetermined at this point is the effective spin-orbit coupling strength $W_{0}$.

Following the simple procedure of Ref. GPHPO16 , in the present work, we set $y_{1}=y_{2}=0$ and assume $\zeta=\zeta^{\prime}$, which leaves only 2 parameters, i.e., $\zeta$ and $W_{0}$, to be determined by nuclear data. In this case, the isoscalar and isovector effective mass parameters, $\mu_{s}\equiv m^{*}/m$ and $\mu_{v}\equiv m_{v}^{*}/m$, where $m$ denotes the nucleon mass in free space, are not independent but are determined via $\zeta$ according to their relations to $\theta_{s}$ and $\theta_{\mu}$ as CBHMS97

[TABLE]

A refined method taking full advantage of the momentum-dependent terms was developed and applied in Ref. GPHO18 . The refinement was found inconsequential for bulk and static nuclear properties. Therefore, the above simplified procedure with $y_{1}=y_{2}=0$ suffices for our present purpose. We now return to the issue of the expansion and examine whether three SNM terms and four PNM terms, a total of seven EoS parameters, are sufficient to achieve convergence of results in the case of nuclei as well as in homogeneous matter.

III Expansion in symmetric part

Equipped with the formalism as discussed above, we now consider the issue of convergence in the description of nuclear properties. The question we address at this stage is how many terms are required for convergence of the expansion in Eq. (1); put in another way, at which order further EoS parameters, such as curvature or compressibility, and skewness, become inconsequential for nuclear applications and thus cannot be constrained by nuclear data. Specifically, we want to know whether higher accuracy can be achieved with more than three terms in SNM and more than four terms in PNM (or the symmetry energy) in practical applications. A negative answer would be of great importance since it would mean that the use of more terms can only lead to overfitting and risk loss of predictive power. The case of SNM will be investigated in this section and the next section is devoted to the case of PNM.

We proceed to examine whether variations in the value of $Q_{0}$ affect strongly the nuclear EoS and the quality of the description of nuclear structure. The empirically determined range of $Q_{0}$ value is between $-1200$ MeV and $-200$ MeV FPT97 , which shows a huge uncertainty. An analysis of nuclear models provides a narrower range $-425.6\sim-362.5$ MeV DLSD12 , which still represents an uncertainty of the order of 15%. Taking this range as a reference, we choose three values of skewness coefficient, $-360$ MeV, $-390$ MeV, and $-420$ MeV. The four parameters $\alpha_{0,1,2,3}$ are now determined by solving a $4\times 4$ system of equations where the coefficients are determined by the assumed values of $\rho_{0},E_{0},K_{0},Q_{0}$.

In the following, the sets of parameters resulting from $Q_{0}=-360$, $-390$, and $-420$ MeV are labeled as S4a, S4b, and S4c, respectively, with the number 4 referring to the number of expansion terms. Presented in Table 1 are the obtained values of the parameters $\alpha_{i}$. In this process, $c_{i}(1)=\alpha_{i}+\beta_{i}$ in Eq. (2) are fixed to the KIDS-ad2 values of Ref. PPLH16 , which parametrize the APR EoS for pure neutron matter. The $N=3$ case, model S3b is obtained with setting $\alpha_{3}=0$ but with $\alpha_{3}+\beta_{3}=956.47$.222For consistency, the model with $N=3$ should be determined with $N=3$ parameterization for PNM. In fact, this model is equivalent to model P3 described in the next section. The final results for S3b and P3 are similar, but, in this section, we work with S3b to vary the SNM parameters only. It can be found that the ranges of $\alpha_{0}$ and $\alpha_{1}$ are rather stable but those of higher order $\alpha_{2,3}$ are sensitive to the input data. Even the signs of the higher order parameters are not robust. This uncertainty is expected because the input data are provided at nuclear saturation density and the higher order coefficients are influenced by higher- and lower-density regions. However, the resulting physical quantities of our interests are not so sensitive as will be shown below.

Figure 1 shows the relative magnitude of each interaction term $\varepsilon_{i}=c_{i}(0)\rho^{1+i/3}=\alpha_{i}\rho^{1+i/3}$, namely, $|\varepsilon_{i}|/\sum_{i}|\varepsilon_{i}|$. The converging behavior $|\varepsilon_{0}|>|\varepsilon_{1}|>|\varepsilon_{2}|>|\varepsilon_{3}|$ is satisfied well up to densities around $3\rho_{0}$ regardless of $N$ or $Q_{0}$ values. At higher densities, where high-order terms are more active, the effects of varying $Q_{0}$ values become clearer as expected. The dominance of the lowest-order term $\varepsilon_{0}$ persists in all cases.

Extrapolation of the model to higher densities is tested by considering properties of the neutron star. It is widely accepted that the core of a neutron star is very asymmetric, so the EoS of a neutron star could be sensitive to the symmetry energy in Eq. (3) that is written in terms of $\beta_{i}$. Since $c_{i}(1)=\alpha_{i}+\beta_{i}$ is fixed by the parameter set KIDS-ad2, but $\alpha_{i}$ varies according to $Q_{0}$ value, $\beta_{i}$ changes to keep $c_{i}(1)$ unchanged, and, consequently, $\mathcal{S}(\rho)$ depends on the $Q_{0}$ value. Figure 2 shows the symmetry energy in various choices of $N$ and $Q_{0}$ values. The dependence on $Q_{0}$ becomes more evident as density increases. However, even around $0.8$ fm*-3* ($\sim 5\rho_{0}$), the maximum difference is only about 20 MeV. The difference becomes appreciable as density reaches about 1 fm*-3*, which is close to the maximum density in the neutron star core and where, in any case, an EDF based on nucleonic degrees of freedom is questionable.

The difference in model predictions on the mass-radius relation of neutron stars is shown in Fig. 3. Predictions for the maximum neutron star mass are 2.11, 2.13, 2.09, and 2.04 $M_{\odot}$, where $M_{\odot}$ is solar mass, for S3b, S4a, S4b, and S4c, respectively. This shows that all the four parameter sets give similar mass-radius properties of neutron stars and allow $2M_{\odot}$ as a neutron star mass. This observation indicates that the effect of fourth order $\varepsilon_{3}$ term and, in particular, the variation of the $Q_{0}$ value within the range of Ref. DLSD12 is marginal in the considered physical quantities.

Now we extend our investigation to the structure of finite nuclei. In order to make use of the Kohn-Sham framework, it is most convenient to transform the EDF to the form of a Skyrme potential, as described in Sec. II.2. When we expand the EDF up to $N=4$, we have $2N$ parameters that are determined from the bulk properties of homogeneous SNM and PNM. However, the effective Skyrme interaction of Eq. (14) has five additional parameters. With the assumption that $y_{1}=y_{2}=0$ and $\zeta=\zeta^{\prime}$, two parameters, $\zeta$ and $W_{0}$, are yet to be determined. The fitting of the undetermined parameters $\zeta$ and $W_{0}$ is performed using 6 data points, namely, the energy per particle ($E/A$) and charge radius ($R_{c}$) of \nuclide[40]Ca \nuclide[48]Ca, and \nuclide[208]Pb. These are listed in the upper 3 rows in Tables 2 and the fitted values of the Skyrme functional parameters are listed in Table 3 for models S3b, S4a, S4b, and S4c. We find that the uncertainties in $c_{i}$ are mostly transferred into those in $t_{32}$ and $t_{33}$, and even their signs change depending on the model. However, the derived physical quantities of the considered nuclei are rather robust. The resulting effective masses $\mu_{s}$ and $\mu_{v}$ of Eq. (18) are obtained as $\mu_{s}=0.99$, 1.03, 0.96, and 0.92, while $\mu_{v}=0.82$, 0.85, 0.79, and 0.77 for S3b, S4a, S4b, and S4c, respectively.We emphasize again that the effective mass values can be allowed to vary, if desired, with no deterioration of the quality of the results on the considered nuclear data. More detailed discussion can be found in Ref. GPHO18 .

The results for \nuclide[16]O, \nuclide[28]O, \nuclide[60]Ca, \nuclide[90]Zr, and \nuclide[132]Sn are also given in Tables 2 for each model. For both $E/A$ and $R_{c}$, fitting quality and predictions of S4a, S4b and S4c are similar, and it is hard to distinguish these models. Furthermore, it is also found that their predictions are similar to those of S3b, which means that the model with $N=3$ is sufficient enough in practical calculations. This result leads to the conclusion that the three leading terms in the isospin symmetric part of the EDF are sufficient to describe not only the bulk properties of neutron stars but also magic nuclei. Both types of systems exhibit the same convergence behavior in a single and unified framework.

IV Expansion in asymmetric part

In this section, we focus on the EDF expansion in asymmetric nuclear matter. We perform this examination by retaining the KIDS-ad2 parameterization ($N=3$) for SNM, which was shown to be sufficient in the description of symmetric matter. With this constraint we proceed to examine the expansion behavior in PNM by varying EoS parameters. In Ref. PPLH16 it was found that at least four terms are needed for satisfactory description of PNM or nuclear symmetry energy. In the present work, we increase the order of expansion of isospin asymmetric part from $N=3$ to $N=6$ and use the APR PNM EoS as input data because of lack of data for PNM. Note that the APR pseudodata are not smooth but show a kink at roughly twice the saturation density. Therefore, as in Ref. PPLH16 , we assign a weight to the data at low energies by defining cost function $\chi^{2}$ as

[TABLE]

where $D_{j}$ is the data point for density $\rho_{j}$, $\mathscr{E}$ and $\mathscr{T}$ are the nuclear EDF and its the kinetic term given by Eq. (1), respectively, and we set $\tilde{\beta}=1$. We refer the details on this form to Ref. PPLH16 .

Fitted values of parameters and the corresponding $\chi_{n}^{2}$ defined as PPLH16

[TABLE]

are shown in Table 4. They are referred to as model P$N$ for $N=(3,4,5,6)$. For $N=6$, we find that there may be more than two sets of parameters that have similar low $\chi^{2}$ values. As examples, we give three sets, P6a, P6b, P6c in Table 4. In particular, P6a is practically equal to P5 and it does not have any physical meaning to work with $N=6$ or higher for APR pseudodata. This is expected since the APR EoS for PNM is determined at densities which can hardly be probed by higher-order terms. The EoS parameters computed for each model are also shown in Table 4. It can be found that although values of model parameters $c_{i}$ would heavily depend on model, the resulting physical quantities or EoS parameters, $J$, $L$, $K_{\rm sym}$, and even $Q_{\rm sym}$ are similar except $R_{\rm sym}$ that depends on the high-order behavior of EDF. We also carry out this kind of analyses with the quantum Monte Carlo (QMC) results of Ref. CGPP14 that are obtained with the AV8’+UIX interaction and verify this observation. The results are presented in Table 5. In this case, we find that there may be more than two sets with similar accuracy even with $N=5$, although we do not list them in Table 5. The fit quality hardly improves in P6. We therefore continue our investigation with models P3, P4, and P5 for the PNM parameters for further exploration.

We first plot the relative magnitudes of individual interaction terms for PNM in Fig. 4 for models P3, P4, and P5. A common aspect in all three cases is the suppression of the $\varepsilon_{0}$ term at high densities. In particular, in P4 and P5, this suppression starts to happen already at the nuclear saturation density as pointed out in Ref. PPLH16 . This behavior is different from that of the SNM case and this would indicates sophisticated dynamics in PNM, which would imply nontrivial isospin dependence of dynamics in nuclear matter and causes huge theoretic uncertainties in nuclear symmetry energy.

Figure 5 shows the energy per particle of PNM for each model and the obtained results are compared to the APR pseudodata. This evidently shows that in order to reproduce the APR pesudodata up to high density region, we need at least $N=4$. And it also shows that $N=5$ does not give any noticeable change from the result of P4.

In Fig. 6, we present the energy per particle of PNM ($E_{\rm PNM}$) divided by the free gas energy ($E_{\rm FG}$) at very low densities. Effective field theory (EFT) results of Ref. DSS13 are presented for comparison by a shaded band. Again, the good agreement with the EFT results is achieved with P4 and higher order terms are irrelevant. The irrelevance of higher order terms of EDF at low densities is not surprising but it is worthwhile to note that the parameters fitted at saturation point can reproduce the results for very dilute system, which is a nontrivial result. Because P4 and P5 have similar EoS, the corresponding neutron star mass-radius curves are expected to be similar and this is confirmed by the results shown in Fig. 7. Here again, the maximum neutron star mass is around $2\,M_{\odot}$.

From the investigation for infinite nuclear matter properties and neutron star mass-radius relations, we conclude that at least four terms are necessary for reasonable descriptions. Then the next question would be whether the parameters determined in this way can describe nuclear properties. Here, we follow the same method and procedure used in Sect. III. The obtained Skyrme parameters are displayed in Table 6, which lead to the binding energy per nucleon and charge radius as presented in Table 7. This shows that there is no significant difference among the predictions of the three parameter sets and even P3 can give a reasonable description of nuclear properties considered in the present work.We also performed these calculations with the parameter sets QMC P3, QMC P4, and QMC P5 listed in Table 5 and they lead to very similar results and conclusions.

As a further test, we repeat the process adopted in Sec. III, namely, we now vary the 4th derivative in nuclear symmetry energy, the kurtosis $R_{\mathrm{sym}}$ in this section. Since the value of $R_{\rm sym}$ obtained from P4 set is about $-2470$ MeV, we consider the variation by $\pm 300$ MeV from this value. For other parameters, we fix $J=32.78$ MeV, $L=49.25$ MeV, $K_{\rm sym}=-156.26$ MeV, and $Q_{\rm sym}=583.07$ MeV. Table 8 presents the values of parameters $c_{i}(1)$ with three different $R_{\rm sym}$ values, which defines P5a, P5b, and P5c. For completeness, we use the values of $c_{i}(0)$ determined as S3b in the previous section.

We first examine the effect of $R_{\rm sym}$ variations on infinite nuclear matter by calculating the neutron-star mass-radius relations. Figure 8 depicts the predictions on the neutron star mass and radius curves with the models P5a, P5b, and P5c. All these models predict the maximum mass larger than $2M_{\odot}$, which is consistent with the observational constraints of Refs. DPRRH10 ; AFWT13 . In order to see the origin of this phenomena, we plot the nuclear symmetry energy for these three models in Fig. 9. This clearly shows that varying $R_{\rm sym}$ affects the nuclear symmetry energy only at densities larger than $0.6\sim 0.7~{}\mbox{fm}^{-3}$ which is within the range of maximum central density of neutron stars LH19 . Since R5a, R5b, and R5c give similar symmetry energy below this density, they would give similar results for neutron stars.

The effect of varying $R_{\rm sym}$ can also be explored in low-mass neutron stars by considering the tidal deformability. For a neutron star with a mass of $1.4M_{\odot}$, we found that P5a, P5b, and P5c models give the dimensionless tidal deformability of 315.8, 304.1, and 289.4, respectively. These values are well below the upper limit of the observation, $\Lambda(1.4M_{\odot})\leq 800$, which again originates from the similarities of symmetry energy of the three models below $0.6\sim 0.7~{}\mbox{fm}^{-3}$.

Tables 9 and 10 show the fitted parameters and resulting properties of nuclei. Here again, we find that the three models give similar results, which leads us to conclude that nuclear properties are quite insensitive to $R_{\rm sym}$. To illustrate the point visually, we compare in Fig. 10 the neutron skin thickness $\Delta r_{np}$ obtained with P3 , P4 (baseline set), P5, P5a, P5b, P5c, together with the results for $E/A$ and $R_{c}$. The similarities shown in Fig. 10 imply that the higher order terms in EDF cannot be constrained by normal nuclear data.333We also investigated the dependence of nuclear properties on the value of $R_{\rm sym}$ by allowing more than $\pm 1,000$ MeV from the value of P5b to confirm that the nuclear properties are not sensitive to $R_{\rm sym}$.

V Discussion

Following the above detailed presentation of results, let us recapitulate what we have done and learned and discuss how our work relates to other current undertakings of similar scope.

First, we have confirmed that seven EoS parameters suffice for a description of nuclei as well as homogeneous matter in a broad range of densities. The number is consistent, on one hand, with the four EoS parameters (plus the surface tension) required in the “minimal nuclear energy functional” BFJPS17 which only concerns finite nuclei; and on the other hand with the conclusions of the recently proposed “meta-modeling” approach for neutron stars MHG17 , namely that the skewness of the EoS plays a non-negligible role, but a less significant one than low-order parameters in the description of neutron stars.

The analytical form of the KIDS EoS and EDF for homogeneous matter, namely an expansion in powers of the cubic root of the density PPLH16 , was inspired by quantum many-body theories and effective field theories. The analytical form allows a straightforward, analytical mapping between the KIDS parameters and an equal number of EoS parameters, see, e.g., Eqs. (9)$-$(LABEL:eq:eosE). Thus we can vary any of the EoS parameters at will and examine its effect on observables. In addition, we are able to vary the effective mass values at will GPHO18 , which gives KIDS unprecedented flexibility. So far we have applied the KIDS EDF at the Hartree-Fock level for nuclear ground states, but studies of excitations within the random phase approximation are also possible and in progress. In this sense our approach goes well beyond the meta-modeling, whose applications in nuclei have been limited to semi-classical results for bulk ground-state nuclear properties CGRM17 ; RG17 .

The description of nuclei was achieved by reverse-engineering a convenient Skyrme-type functional. In the process, the amount of momentum dependence (encoded, for example, in the effective mass value and gradient terms) vs. genuine density dependence (encoding correlations and three-nucleon forces) needs to be determined. Although we have found that static, bulk nuclear properties are practically independent of the effective mass GPHO18 , the same may not be true for dynamic phenomena such as giant resonances. Studies are in progress PG18 . Nevertheless, the small amount of momentum relative to density dependence generally favored by our studies so far, undermines the possibility to eliminate density-dependent couplings completely, as is attempted in certain generalizations of the Skyrme functional based on high-order momentum-dependent terms and on the density-matrix expansion CDK08 ; DPN15 .

Based on our present results we may conclude that a fit of more than the above seven EoS parameters to nuclear data would make little sense. (On the contrary, a free fit of all parameters could lead to overfitting.) Although further EoS parameters and a strong momentum dependence are not desired or required, in order to achieve precision, it does make sense to explore extensions of the KIDS EDF for nuclei by including additional effects which are not active (or are weakly active) in homogeneous matter. One of them, already included, is the spin-orbit term. Another interesting possibility is the tensor force, as already pursued in modern Skyrme functionals LBBDM07 . Time-odd terms are also unconstrained at present. Our preferred approach would be to use pseudodata for polarized homogeneous matter.

VI Summary and conclusion

The main purpose of this work was to validate the optimal number of EoS parameters required for a description of nuclei and homogeneous matter in a broad range of densities. Previous work in the framework of the KIDS EDF had indicated that symmetric nuclear matter could be efficiently modeled with three low-order parameters in an expansion in Fermi momentum and that PNM requires four parameters. The conclusion was based solely on a statistical analysis of fits to pseudodata for homogeneous matter. In this work, in order to confirm the expansion and its convergence, we explored the role of widely used parameters characterizing the EoS at the saturation point. In particular, we fixed the saturation density, the energy at saturation and the compression modulus $K_{0}$ of symmetric matter, as well as the symmetry energy at saturation density $J$, its slope $L$ and its curvature and skewness, to baseline values and varied the EoS skewness in symmetric matter at saturation, $Q_{0}$, and the kurtosis of the symmetry energy, $R_{\mathrm{sym}}$. We examined the effect in dilute and dense matter (neutron star properties) and on nuclear structure.

In regard to the uncertainty from $Q_{0}$, its effect is negligible up to $\rho\sim 0.4\,{\rm fm}^{-3}$ ($\sim 2.5\rho_{0}$). The maximum mass of neutron stars shows non-negligible dependence on $Q_{0}$, but the uncertainty is not significant enough to affect the consistency with existing observations. No effect on bulk nuclear properties was discerned.

In the extension of expansion of isospin asymmetric part of EDF, the results for $N=6$ showed symptoms of overfitting so we stopped at the fifth term. Comparison of $N=3$ fitting result to input data demonstrated that three terms in asymmetric part are insufficient to guarantee the reproduction of input data but the fits saturate at $N=5$. The interpretation is consistent with the EoS of dilute neutron matter, symmetry energy at high densities, and mass-radius curves of neutron stars. Again, the choice of kurtosis values $R_{\mathrm{sym}}$ did not affect the description of nuclear properties.

Bulk properties of spherical magic nuclei were calculated. Results turned out to be independent of $Q_{0}$ values, and the number of terms in asymmetric part of EDF did not affect the prediction for nuclei. Similar conclusions hold for $R_{\mathrm{sym}}$.

From the present results we conclude that three terms in the symmetric part, and four terms in the asymmetric part of the EoS are sufficient for a unified description of both infinite (unpolarized) nuclear matter and finite nuclei in a single framework. Fitting a nuclear EDF with more than the seven necessary EoS parameters to nuclear data can arguably lead to overtraining and loss of predictive power. The determination of the most realistic values for the minimal EoS parameters can of course be persued with the help of data and statistical analyses. In addition, extended density dependencies of non-local terms can be explored PF94 ; FPT97 ; FPT01 . The EoS of polarized matter is yet another topic to be considered. But attempts at refining the nuclear EDF beyond that number of terms must focus on parameters which are not active (or strongly active) in unpolarized homogeneous matter, for example the effective tensor and time-odd terms.

Acknowledgments

Y.O. is grateful to Yeunhwan Lim for useful information and comments. This work was supported by the National Research Foundation of Korea under Grants Nos. NRF-2017R1D1A1B03029020, NRF-2018R1D1A1B07048183, NRF-2018R1A6A1A06024970, and NRF-2018R1A5A1025563. The work of P.P. was supported by the Rare Isotope Science Project of the Institute for Basic Science funded by Ministry of Science and ICT (MSIT) and the National Research Foundation of Korea (No. 2013M7A1A1075764).